Notes on Chapter 3: Elementary Quantum Physics

Overview of Elementary Quantum Physics

  • 3.1 Photons
  • 3.2 The Electron as a Wave
  • 3.3 Infinite Potential Well: A Confined Electron
  • 3.4 Heisenberg's Uncertainty Principle
  • 3.5 Confined Electron in a Finite Potential Energy Well
  • 3.6 Tunneling Phenomenon: Quantum Leak
  • 3.7 Potential Box: Three Quantum Numbers
  • 3.8 Hydrogenic Atom
  • 3.9 The Helium Atom and the Periodic Table
  • 3.10 Stimulated Emission and Lasers
  • 3.11 Optical Fiber Amplifiers

Wave-Particle Duality

  • Electrons and Light:
    • Electrons are typically described as particles obeying Newton's laws (e.g., F = ma) but also exhibit wave-like properties.
    • Diffraction and interference patterns can be produced with an electron beam, similar to light.
    • Light is typically treated as an electromagnetic wave with particle-like behaviors (photons).

Light as a Wave

  • Properties and Behaviors:
    • Light exhibits wave phenomena such as interference, diffraction, refraction, and reflection.
    • Described as an electromagnetic (EM) wave.
    • Wave Equation: Electric field Ey(x,t) = E0 ext{sin}(kx - wt); where:
    • k = rac{2 ext{π}}{ ext{λ}} (wavenumber)
    • ω = 2 ext{π}f (angular frequency)
  • Intensity Calculation:
    • I = rac{1}{2}c ext{ε}0E0^2, where ext{ε}_0 is the absolute permittivity.

Light as an Electromagnetic Wave

  • Visualization: Electric and magnetic fields are mutually perpendicular and also perpendicular to the direction of wave propagation.

Young’s Double Slit Experiment

  • Interference Patterns:
    • Bright and dark fringes are created based on the phase difference between waves passing through slits:
    • Constructive interference: S1P - S2P = n ext{λ}
    • Destructive interference: S1P - S2P = rac{n + rac{1}{2}}{ ext{λ}}
  • Bragg’s Law for X-rays:
    • 2d ext{sin} heta = n ext{λ} describes the condition for constructive interference in X-ray diffraction.

Light as a Particle

  • Photons:
    • Light can be viewed as a stream of photons, with:
    • Energy: E = hf
    • Momentum: p = rac{h}{λ}

The Photoelectric Effect

  • Experiment Setup:
    • Involves illuminating a cathode with light to emit electrons, monitored by an ammeter.
  • Key Observations:
    • The current I is dependent on light frequency, with a threshold frequency f_0 past which electron emission occurs.
    • Kinetic energy of emitted electrons: KEm = hf - hf0.
    • The effect demonstrates that light's intensity affects only the number of emitted electrons, not their kinetic energy.

Compton Scattering

  • Interaction of X-ray with Electrons:
    • An X-ray photon colliding with an electron can transfer energy, resulting in a lower frequency photon post-collision:
    • The scattered electron gains kinetic energy.
  • Momentum Conservation:
    • Scattering demonstrates the particle-like nature of photons, with momentum defined as p = rac{h}{λ}.

The Electron as a Wave

  • Experimental Evidence:
    • Interference and diffraction observed with electron beams supports wave-particle duality for electrons.
  • De Broglie Hypothesis:
    • The wavelength of a particle is given by λ = rac{h}{p}, showing connections between wave and particle properties.

Exercises

  1. Photon Energy Calculation: Determine the energy of a blue photon with a wavelength of 450 nm.
  2. Photoelectric Experiment Analysis: Calculate the work function of sodium exposed to photons.
  3. Momentum and Energy Calculations: Work out values for X-ray photons and corresponding electron momentum.

Constants and Important Values

  • Plank’s constant: h = 6.626 × 10^{-34} J ext{s}
  • Shorthand Definitions: 1 ext{ eV} = 1.6 × 10^{-19} J
  • Electron properties: Mass m_e = 9.109 × 10^{-31} ext{ kg}, Charge e = 1.6 × 10^{-19} C.